A Generalized Walsh System and Its Fast Algorithm

Abstract

In this paper, we propose a new class of discontinuous orthogonal system called Walsh-U system, which is composed of piecewise polynomials of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> in Hilbert space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}[0, 1]$</tex-math></inline-formula> . The Walsh-U system generalizes the Walsh system from piecewise constant to piecewise polynomials, which is capable of representing discontinuous signals. Besides, the basis functions of the Walsh-U system not only possess a more concise construction, but also have stronger sparse representation capabilities than the U-system. In the construction of the Walsh-U system, the Legendre polynomials are adopted as the generators at first, and then a series of basis functions are produced through the generators and basic operations of scaling and replication. These basis functions are formed into different groups based on the generation order. Furthermore, a fast algorithm for the Walsh-U system is designed in this paper. The experimental results demonstrate that when compared with Fourier series, Walsh functions, wavelet functions, and U-system, the Walsh-U system has fast convergence and high approximation precision in analog signal reconstruction and denoising. It is verified that the fast algorithm for the Walsh-U system can effectively improve the computation speed for discontinuous signals.